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Toroidal compacifications and incompressibility of exceptional congruence covers
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Patrick Brosnan
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ABSTRACT
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Suppose a finite group G acts faithfully on an irreducible
variety X. We say that the G-variety X is compressible if there is a
dominant rational morphism from X to a faithful G-variety Y of
strictly smaller dimension. Otherwise we say that X is
incompressible. In a recent preprint, Farb, Kisin and Wolfson (FKW)
have proved the incompressibility of a large class of covers related
to the moduli space of principally polarized abelian varieties with
level structure. Their methods, which rely on the existence of
integral models for the moduli space Ag, apply to diverse examples
such as moduli spaces of curves and many Shimura varieties of Hodge
type. My talk will be about joint work with Fakhruddin and
Reichstein, where our goal is to recover some of the results of FKW
via the fixed point method from the theory of essential dimension.
More specifically, we prove incompressibilty for some Shimura
varieties by finding fixed points of finite abelian subgroups of G in
their toroidal compactifications. Our results are weaker than the
results of FKW for Hodge type Shimura varieties, because the methods
of FKW apply in cases where there is no boundary, while we need the
boundary to find the fixed points. However, our method has the
advantage of extending to many Shimura varieties which are not of
Hodge type, in particular, those associated to groups of type
E7. Moreover, by using Pink's extension of the Ash, Mumford, Rapoport
and Tai theory of toroidal compactifications to mixed Shimura
varieties, we are able to prove incompressibility for congruence
covers corresponding to certain universal families: e.g., the universal
families of principally polarized abelian varieties.
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